GEOLOGICAL CHANGES ON THE EARTH’S AXIS OE ROTATION. 
:01 
Now the propositions given in Part II., as to the change in the position of the earth’s 
axis, remain true here also ; and the only difference is that the products of inertia D 
and]E must now be expressed by different integrals. 
After intumescence the earth may be conceived to consist of : — -first, itself as it was 
before; secondly , of negative matter, of which the law of density is Q^r~, throughout 
the space bounded by r=r x , r=c, and the cone of elevation ; and, thirdly , of the 
matter which formerly lay within this space, in the configuration attained by it after 
intumescence. 
The first part clearly contributes nothing to D and E ; and the second contributes 
— Q JVfr 3 sin qr sin 2 9 cos 9 ^ dr d9 dq>, 
integrated throughout the above space, that is from r=r 1 to r—c, and throughout the 
cone of elevation. 
As to the third part, the mass of any element remains unchanged, whilst its distance 
from the centre has become r-\-u. Hence the third part contributes 
Q jjj r{r-{-uY sin qr sin 2 9 cos 9 j ^ ^ dr d9 d<p, 
integrated throughout the above space. 
Therefore, taking all together, and treating u as small, 
^ 1 =2Q (TV ur 2 sin qr sin 2 9 cos 9 ( sin ^ 
LJ JJJ * [cos<p 
Therefore 
dr d9 d<p. 
^=“=2Q fwr 2 sin qr dr, 
where d and e have the same meanings as before, in Part II. sec. 12. 
Now this last integral divides itself into two parts: first, from r=c to r=r 2 , u—h ; 
Cl y 3 
and, secondly, from r=r 2 to r=r„ u=^ . — 
Therefore 
^j-=^=2Q/j J* r 2 singr dr+§Qcc^ (r 3 —r 3 ) sin qr dr. 
If the value of a be substituted, and the integrations effected, it will be found that 
D _ E _ 1 2 S_ 3c 1 1 T 
2d Q,h<? 2eQ/«c 3 qc cos qc '(gc) 2 sin qc' r 2 ’ (g- c ) 2 sin qc suppose, 
where S stands for the expression r - sin qr -j- cos qr, taken between the limits c and r, 2 , 
and T for the expression 
r 2 and r v 
© 
sin qr 
qc 
cos qr- 
[qcy 
sin qr, taken between the limits 
